Limit rotation to 180+ in X axis?
As posted on BlenderArtists, here's the solution as provided below:

I simply added a limit rotation constraint to a duplicate of the main object, which I shrunk down (not necessary) and added a "limit rotation constraint" which I set to limit the X axis rotation to -30 degrees. 

Then, I added a driver (right-click the desired value in the sidebar to the right in the 3D view and select "Add Driver") to the X axis for the "Axe_motor" (the main object) and set the driver's expression be "var * 6.26" in my case. 

So, I multiplied -30 degrees by 6.26 times. this resulted in a perfect arc from the axe's resting position (0 degrees) to -188 degrees in X.